# Count of all combinations of a set

All right, sorry in advance if this exists on here already. What I'm after is figuring out all possible orders of a set.

So I have 2 items

1,2
2,1
=2


3 items

1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1
=6


I can't seem to find a correlation I can burn down to an equation.

Any ideas? Thanks.

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go nowhere --> en.wikipedia.org/wiki/Permutation –  jay-sun Feb 28 '13 at 21:46

For computing the number of permutations of $n$ elements of a set, (distinct arrangements of elements of the set), there are $$n! = n(n-1)(n-2) \cdots 2 \cdot 1$$ possible combinations.

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So for $n = 2, 2! = 2 \times 1 = 2$ permutations. For $n = 3,$ there are $3! = 3\times 2 \times 1 = 6$ permutations. For n = 5, there are $5! = 5 \times 4\times 3\times 2 \times 1 = 120$ possible permutations of a set of size $5$. $n!$ reads $n$-factorial: see factorial –  amWhy Feb 28 '13 at 21:56
The logic makes sense when you consider that for a set of, say $4$ elements, there are $4$ choices for picking which element is first, then $3$ choices for picking which element is second, then only $2$ choices for picking which element is third, and we're stuck with only one option for the last element. So there are $4 \times 3 \times 2\times 1 = 4!$ possible unique combinations/permutations. –  amWhy Feb 28 '13 at 22:42

for $\emptyset$ with $0$ elements we have 1 permutation

for $\{1\}$ set with $1$ element we have $$1$$ permutation

for $\{1,2\}$ set with $2$ elements we have $$2=1\cdot2=2!$$ permutations

$12$

$21$

for $\{1,2,3\}$ set with $3$ elements we have $$6=1\cdot2\cdot3=3!$$ permutations

$123,132$

$213,231$

$312,321$

for $\{1,2,3,4\}$ set with $4$ elements we have $$24=1\cdot2\cdot3\cdot4=4!$$ permutations

$1234,1243,1324,1342,1423,1432$

$2134,2143,2314,2341,2413,2431$

$3124,3142,3214,3241,3412,3421$

$4123,4132,4213,4231,41312,4321$

Continuing this way we have that set with n elements has $$1\cdot 2\cdot3\cdot...\cdot n=n!$$ permutations

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+1 long nice illustration Adi. :) –  B. S. Mar 1 '13 at 6:37
+1 for getting 0! out of the way. –  Loki Clock Mar 2 '13 at 7:00

So you mean ordered combinations that use all the elements in every combination. Since a set can't contain the same element more than once, you're talking about the number of permutations of the set, which is n!, where n is the size of the set.

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I'm sorry, I don't really know the lingo, I also am not sure what n! is. –  kelton52 Feb 28 '13 at 21:48
Sorry, fixed the definition of n. –  Loki Clock Feb 28 '13 at 21:49
So with a set of 2 2(2-1)(2-2)(2-2+1)=0? Do you just not add the last 2-2? –  kelton52 Feb 28 '13 at 21:54
You start from 1 and then multiply by 2, etc.. So you got 6 because 1*2*3=6. –  Loki Clock Feb 28 '13 at 22:34

The classroom thinking is : I have n boxes

How many elements can i can choose to put in the first box : $n$

How many elements can i can choose to put in the second box : $n - 1$ since i have put one in the first box.

And so on, so the total posibilities are $n \times n-1 \times \dots 1 = n!$, the factorial number.

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$n! = n \cdot (n-1) \dots 1$

In C/C++:

int max = n ; //( 3 or 2 .. )
total = max;

while (max > 0){
max --;
total = total * max;
}

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Can you add one more ... in there? I can't quite see the pattern. –  kelton52 Feb 28 '13 at 21:50
I believe you can see it now since you are a developer –  Mr.Me Feb 28 '13 at 21:59
Heh, thanks. That does help. –  kelton52 Feb 28 '13 at 21:59
Well actually, wouldn't this be better? int setSize; int sampleSize; int result = setSize; for(int i=setSize;i>0;--i) { result = result * i; } result = result * ((setSize - sampleSize) + 1); –  kelton52 Feb 28 '13 at 22:03